Hypocycloid Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/HypocycloidPedalCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HypocycloidPedalCurve.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

PedalCurve[Hypocycloid[{a,b},t],{x0,y0},t]//FullSimplify

TraditionalForm[%/.{x0->(Subscript[x, 0]),y0->(Subscript[y, 0])}]

PedalCurve[Hypocycloid[{a,a/n},t],{x0,y0},t]//FullSimplify

TraditionalForm[%/.{x0->(Subscript[x, 0]),y0->(Subscript[y, 0])}]

At Origin

Hypocycloid[{a,a/n},t]//Simplify//TraditionalForm

PedalCurve[Hypocycloid[{a,a/n},t],{0,0},t]//FullSimplify

%//TraditionalForm

Table[{(( (-2+n) (Cos[t]-Cos[t-n t]))/(2 n)),(( (-2+n) Cos[t-((n t)/2)] Sin[((n t)/2)])/n)},{n,3,6}]//FullSimplify

ped=PedalCurve[Hypocycloid[{n,1},t],{0,0},t]//FullSimplify

FullSimplify[Sqrt[ped.ped],0<t<2π&&n>1]

FullSimplify[ArcTan[Divide@@Reverse[ped]]]

Solve[θ==-ArcTan[Cot[t-((n t)/2)]],t]

Sin[((n t)/2)]/.t->((2 ArcCot[Tan[θ]])/(-2+n))

Plot

<<Utilities`Typesetting`

Off[Power::"infy",∞::"indet",Graphics::"gptn"];

Show[GraphicsArray[Partition[
Block[{$DisplayFunction=Identity},Table[PedalCurvePlot[Hypocycloid[{n,1},t],{0,0},{t,0,2Pi},{0,2Pi/n,Pi/n/5}],{n,3,8}]
],3]]]

Show[GraphicsArray[
Partition[Table[PolarPlot[Sin[(n/(n-2))(t+Pi/2)],{t,0,(n-2)π },PlotStyle->Red,PlotLabel->TraditionalForm[HoldForm[n]==n],DisplayFunction->Identity],{n,3,8}],3]
]]